The conjecture of Sally on the Hilbert function for curve singularities
type: Item
instance of: scholarly article
title: The conjecture of Sally on the Hilbert function for curve singularities
zbMATH Open document ID: 0799.13007
author: J. Elias
published in: Journal of Algebra
publication date: 1994-11-16T00:00:00Z
review text: Let \(A\) be a one-dimensional Cohen-Macaulay local ring with maximal ideal \({\mathfrak m}\). \textit{J. Sally} [in ``Numbers of generators of ideals in local rings'', Lect. Notes Pure Appl. Math. 35 (1978; Zbl 0395.13010)] conjectured that if \(A\) is of small embedding dimension, then the Hilbert function length\(({\mathfrak m}^ n/{\mathfrak m}^{n + 1})\) is nondecreasing. \textit{F. Orecchia} [Manuscr. Math. 32, 391-405 (1980; Zbl 0445.13008)] proved that there exist reduced one-dimensional local rings of embedding dimension \(\geq 5\) with decreasing Hilbert function. In this paper, the conjecture is settled in the affirmative for equicharacteristic rings of embedding dimension 3. The key point in the proof is the fact that \(A\) is defined by a codimension 2 ideal so that one can use the Hilbert-Burch theorem.
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 436479
zbMATH Keywords: non-decreasing Hilbert function
zbMATH Keywords: Cohen-Macaulay local ring
zbMATH Keywords: embedding dimension
Wikidata QID: Q122944911
reviewed by: Trung Ngo Viet
MaRDI profile type: Publication
full work available at URL: jabr.1993.1176
OpenAlex ID: W2019312990
DOI: 10.1006/JABR.1993.1176
description: scientific article; zbMATH DE number 436479